A ferry is crossing a river. The ferry is
headed due north with a speed of 2.4 m/s
relative to the water and the river’s velocity
is 2.3 m/s to the east.
a) What is magnitude of the boat’s velocity
b) Find the direction in which the ferry is
moving (measured from due east, with counterclockwise positive).
Answer in units of
◦
For A you need to find the resultant. So since you know it moves 2.4 m/s north and 2.3 m/s east you can use the Pythagorean theorem to find it.
For B i am not as sure about but it most likely is asking you to find the angle (based from east) where the boat is moving. So use inverse tan (opp/adj)
To find the magnitude of the boat's velocity, we can use the concept of vector addition.
The magnitude of the boat's velocity can be found using the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the magnitude of the boat's velocity (Vboat) is the hypotenuse, while the boat's velocity relative to the water (2.4 m/s) is one side and the river's velocity (2.3 m/s to the east) is the other side.
Using the Pythagorean theorem, we have:
Vboat^2 = (2.4 m/s)^2 + (2.3 m/s)^2
Vboat^2 = 5.76 m^2/s^2 + 5.29 m^2/s^2
Vboat^2 = 11.05 m^2/s^2
Taking the square root of both sides, we find:
Vboat = √11.05 m/s
Therefore, the magnitude of the boat's velocity is approximately 3.32 m/s.
To find the direction in which the ferry is moving, we can use the trigonometric function tangent. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the river's velocity (2.3 m/s to the east) and the adjacent side is the boat's velocity relative to the water (2.4 m/s).
Using the tangent function, we have:
tan(θ) = opposite/adjacent
tan(θ) = 2.3 m/s / 2.4 m/s
Simplifying the right side, we find:
tan(θ) ≈ 0.9583
To find the value of θ (angle), we can take the inverse tangent (also known as arctan) of both sides:
θ ≈ arctan(0.9583)
Using a calculator, we find:
θ ≈ 43.68°
Therefore, the direction in which the ferry is moving, measured from due east with counterclockwise positive, is approximately 43.68°.